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Consider a cost function that is continuous, differentiable and (possibly) convex: $c(x):\mathbb{R}^+\to \mathbb{R}$. I was wondering if there is a "common" way to interpret the expression:

$$xc'(x)$$

in terms of economic intuitions. Any suggestions or references?

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If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem,

$$ x c'(x) = k c(x).$$

That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is homogeneous of degree 1, then $x c'(x) = ax = c(x)$).

Without homogeneity, I am not sure there is any insightful interpretation you can give to $x c'(x)$, but others may have better ideas. The reason I am pessimistic is the "most natural" way for $x c'(x)$ to appear is in

$$[xc(x)]' = xc'(x) + c(x),$$

which gives

$$\frac{[xc(x)]'}{c(x)} = xc'(x).$$

However, it's not clear to me how to interpret $\frac{[xc(x)]'}{c(x)}$ either. Why should anyone care about $xc(x)$ in the first place? Provided $c(x)$ is the total cost (and not the per unit cost), why would you want multiply your total cost by the number of units you are interested in (what does "the total cost of producing 5 units times 5" means?).

If you let $c(x)$ be the per unit cost, maybe you can make some interpretative sense of $\frac{[xc(x)]'}{c(x)}$ as $xc(x)$ then becomes the total cost?

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